From 6e475df77ad66c36ac3405cb553f5de0e64a94e5 Mon Sep 17 00:00:00 2001 From: zleyyij <75810274+zleyyij@users.noreply.github.com> Date: Wed, 18 Sep 2024 12:12:12 -0600 Subject: [PATCH] vault backup: 2024-09-18 12:12:12 --- education/math/MATH1060 (trig)/Identities.md | 20 ++++++++++++++------ 1 file changed, 14 insertions(+), 6 deletions(-) diff --git a/education/math/MATH1060 (trig)/Identities.md b/education/math/MATH1060 (trig)/Identities.md index 1382000..827001f 100644 --- a/education/math/MATH1060 (trig)/Identities.md +++ b/education/math/MATH1060 (trig)/Identities.md @@ -5,15 +5,23 @@ All of the following only apply when the denominator is not equal to zero. $$ tan \theta = \frac{y}{x} $$ Because the following are inverses of their counterparts, you only need to remember the equivalents for $sin$, $cos$, and $tan$, then just find the inverse by taking $1/v$. -| Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities | -| ----------------------------- | ------------------------------ | --------------------------------------------- | ------------------------------------------------------------------------- | -| $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ | -| $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | | -| $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ |
$$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ | -| | | | | +| Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities | +| ----------------------------- | ------------------------------ | --------------------------------------------- | --------------------------------------------------------------------- | +| $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ | +| $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | $$ sec\theta = \frac{1}{cos\theta} $$ | +| $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ | $$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ | + $$ cot \theta = \frac{x}{y} $$ $$ sec\theta = \frac{1}{cos\theta}$$ $$ csc\theta = \frac{1}{sin\theta}$$ # Pythagorean Identities The Pythagorean identity expresses the Pythagorean theorem in terms of trigonometric functions. It's a basic relation between the sine and cosine functions. $$ sin^2 \theta + cos^2 \theta = 1 $$ +There are more forms that are useful, but they can be derived from the above formula: +$$ 1 + tan^2\theta = sec^2\theta $$ +$$ cot^2 \theta + 1 = csc^2\theta $$ +# Even and Odd Identities +- A function is even if $f(-x) = f(x)$. +- A function is odd if $f(-x) = -f(x)$ +Cosine and secant are **even* +## Examples